A hyperbolic Kac–Moody Calogero model

A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac–Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra AE 3 with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincar´e disk with a unique potential. Since the Weyl group of AE 3 is a Z 2 extension of the modular group PSL(2, Z ), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of AE 3 , give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincar´e series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case we find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.


Introduction
To every finite Coxeter group of rank n one can associate a (classical and quantum) maximally superintegrable mechanical system known as a (rational) Calogero-Moser model (Calogero model for short) living in 2n-dimensional phase space with momenta p i and coordinates x i collected into x P R n [1,2,3].It is determined by the Hamiltonian where '¨' denotes the standard Euclidean scalar product in R n , and the sum runs over the root system R consisting of all nonzero roots α belonging to the Coxeter-group reflections1 s α : R n Ñ R n via s α x " x ´2 x¨α α¨α α . (1. 2) The real coupling constants g α are constant on each Weyl-group orbit, so in an irreducible simply-laced case they all agree, g α "g.For the classical Hamiltonian, ℏ"0, while in the quantum case we represent the momenta by differential operators, Bx i ": ´iℏ B i such that rx i , p j s " iℏ δ i j . (1.3) Henceforth we set ℏ " 1 for convenience.
There exist a variety of generalizations for these models, but we like to mention only their restriction to the unit sphere S n´1 given by x¨x " 1, which has been named the (spherical) angular Calogero model [4,5,6].In the quantum version, L2 is the (scalar) Laplacian on S n´1 , and the potential U depends only on its angular coordinates ϑ " tϑ 1 , . . ., ϑ n´1 u.Since V pxq is singular at the mirror hyperplanes α¨x"0 of the Coxeter group, U pϑq blows up at their intersection with the unit sphere.The full as well as the angular Calogero model and their generalization have a rich history as paradigmatic many-body integrable models (for a review, see [7,8]).
Generalizing to infinite Coxeter groups of the affine type renders the coordinates periodic, x P T n , which turns rational Calogero models into Sutherland models [9].However, to the author's knowledge, hyperbolic Coxeter groups [10] have not been employed to this purpose, except for [11] on Toda field theories and [12] 2 on a more general class of Calogero models.
In the present paper, we propose a rational Calogero model based on one of the simplest hyperbolic Coxeter groups, namely the paracompact right triangular hyperbolic group labelled by rp, q, rs " r2, 3, 8s and Coxeter-Dynkin diagram ‚-8 ‚-3 ‚ (see Figure 1).This happens to be the Weyl group of the simplest hyperbolic rank-3 Lie algebra AE 3 , a double extension of A 1 " sl 2 [13]. 3Its root space is of Lorentzian signature, which we take as p´, `, `q.Denoting the phase-space coordinates by px µ , p µ q for µ " 0, 1, 2, the Minkowski metric by η µν and the Lorentzian scalar product again by '¨', the Hamiltonian then has the form (employing the Einstein summation convention and pulling the coupling out of the potential) where the sum is restricted to the set R of real roots, which we normalize to α¨α " 2. Since R for AE 3 decomposes into two Weyl orbits R `and R ´ [19], we may actually split the potential into two pieces and weigh them individually.However, for the sake of simplicity we keep the couplings equal for this paper, g `"g ´"g.We also do not consider the possible inclusion of imaginary roots in the potential.Like the Euclidean theory can be reduced to the unit sphere, the Minkowskian variant can be restricted to the one-sheeted hyperboloid (x¨x " 1) or (one sheet of) the two-sheeted hyperboloid (x¨x " ´1).In order to produce a model on a Riemannian manifold (with Euclidean signature), we consider the future hyperboloid H 2 given by x¨x " ´1 and x 0 ą 1.Let us parametrize the Minkowski future by x 0 " r cosh θ , x 1 " r sinh θ cos ϕ , x 2 " r sinh θ sin ϕ with r P R ą0 , θ P R ě0 , ϕ P r0, 2πs (1.6) so that we may restrict to r"1 and obtain the quantum Hamiltonian of a "hyperbolic Calogero model"4 ) where L 2 is just the (scalar) Laplacian on H 2 .Our task will be to compute and characterize the potential V respective U .

The real roots of the Kac-Moody algebra AE 3
In order to formulate the Calogero potential for the real roots of AE 3 we need to collect some facts about this simplest of hyperbolic Kac-Moody algebras [15,14,13].Starting from its Cartan matrix, we parametrize the three simple roots α µ of length-square 2 in three-dimensional Minkowski space R 1,2 with a Minkowski-orthonormal basis te µ u " te 0 , e For symmetry reasons we add the non-simple root so that the overextended simple root can be rewritten as (2.5) The three roots α i , i " 1, 2, 3, belong to an A 2 subalgebra and obey the relations The real roots of AE 3 lie on the one-sheeted hyperboloid x¨x"2 and are given by where the length condition translates to the diophantine equation pℓ ´mq 2 `n pn ´mq " 1 . (2.8) Since the roots come in pairs pα, ´αq, it suffices to analyze ℓ ě 0 only.At any given "level" ℓ the solutions furnish (generically several) highest weights of the "horizontal" A 2 " sl 3 subalgebra plus their images under its S 3 Weyl-group action, pm, nq Ñ pm, m´nq Ñ p2ℓ´n, m´nq Ñ p2ℓ´n, 2ℓ´mq Ñ p2ℓ´m`n, 2ℓ´mq Ñ p2ℓ´m`n, nq .
(2.9)Such a sextet of weights belongs to an A 2 representation with Dynkin labels rp, qs " r´2ℓ`2m´n, ´m`2ns ô pm, nq hw " 1 3 p4ℓ`2p`q, 2ℓ`p`2qq (2.10) for the highest-weight values pm, nq hw in (2.9). Figure 2 shows the distribution of the real roots for low levels.We obtain a parametrization more symmetric under spatial rotations by replacing α 0 with e 0 using (2.5) and employing a kind of barycentric coordinates, x " x 0 e 0 `x i α i with x1 `x 2 `x 3 " 0 . (2.11) In these coordinates, the real roots take the form α " b 2 3 ℓ e 0 `ᾱ with ᾱ " 1 3 p α 1 `1 3 q α 2 `1 3 r α 3 and p `q `r " 0 , (2.12) where p and q coincide with the Dynkin labels in case ᾱ is a highest A 2 weight.The S 3 Weyl group action simply permutes the coefficients pp, q, rq and multiplies them with the sign of the permutation.On a given level ℓ the A 2 weights ᾱ all have the same length-square ᾱ¨ᾱ " 2 `2 3 ℓ 2 .On may translate the diophantine equation (2.8) to the Dynkin labels and obtain ´pq ´pr ´r p " p2 `q 2 `p q " ℓ 2 `3 as well as 3 | ℓ ´p `q . (2.13) The number of A 2 representations grows erratically with the level, as displayed in Figure 3. Two representations appear first at ℓ"6, three at ℓ"12, four at ℓ"30, eight at ℓ"72 and so on.Although the number of solutions grows quickly with ℓ it is easy to give a few infinite families of real roots (see also [13]),  (2.22) Each such hyperbolic reflection preserves the radial coordinate r " ?´x¨x and the time orientation but reverses the spatial orientation (det S " ´1), hence it represents an involution on the future hyperboloid.A collection of such mirrors is displayed in Figure 5, and Figure 6 shows their intersections with the x 0 "1 plane.

The potential
The "horizontal" A 2 slicing of the root space into levels ℓ P Z leads to a decomposition of the potential, where R ℓ denotes the set of real roots α with α¨e 0 " b 2 3 ℓ.Clearly, V ´ℓ " V ℓ .Let us take a look at levels zero and one.Summing over the adjoint representation of A 2 , V 0 pr, θ, ϕq " is just the celebrated Pöschl-Teller potential, modulated by a θ dependence.For level one we sum over the three extremal weights of the r0, 2s representation, It is also possible to sum over whole families of solutions to the diophantine equation (2.8).Let us do so for the first family in (2.15), extending it to negative levels (to include the negative real roots) and including levels zero and one with their proper weight inside V , where the V 0 contribution got cancelled on the way.Although this is only a part of the full potential, it does show some characteristic features of V : • the A 2 subgroup's Weyl group S 3 yields a dihedral symmetry and six-fold mirrors, • infinitely many mirrors intersect in the three null lines λn i , • in the ℓ Ñ 8 limit the mirrors accumulate in the three null planes n i ¨x " 0.
• the mirrors tessalate the interior of the lightcone with infinitely many triangular Weyl alcoves • a fundamental Weyl alcove is spanned by te 0 , 2e 0 `e2 , 2e 0 `?3e 1 `e2 u A contour plot of log V on the plane x 0 "1 is given in Figure 7 for one Weyl alcove.

Mapping to the complex half-plane
Since our model is scale invariant, for the potential we can restrict ourselves to the future hyperboloid r 2 " ´x¨x " 1 and x 0 ě 1.It is convenient to pass to complex embedding coordinates t " x 0 and w " x 1 `ix 2 with t 2 ´ww " 1 .
By a stereographic projection (see Figure 8) the hyperboloid gets mapped to the unit disk vv ď 1 for The metric induced from the Minkowski metric turns this into the Poincaré disk model of the hyperbolic where ρ " e 2πi{3 ñ ρ 2 " ρ , ρ `ρ " ´1 , 1 `ρ `ρ2 " 0 .(4.4) Adding the infinity of real-root mirrors produces a paracompact triangular tessalation of type rp, q, rs " r2, 3, 8s.Each of the hyperbolically congruent triangles has angles π 2 , π 3 and 0, and at the corresponding vertices there meet 4, 6 and infinitely many triangles, thus one vertex is always at the boundary, as is visible from Figure 9.We employ a variant of the Cayley map to further pass to the complex upper half plane such that the boundary |v|"1 becomes the real axis Im z"0.The direct relation between w and z reads and the mirror curves at level zero and one become (in the same order as in (4.3)) ℓ " 0 : The first two curves in the ℓ"0 list and the first one at ℓ"1 bound a standard fundamental domain which is co-finite (with a hyperbolic volume of π 6 ) but not co-compact due to a cusp at i8. Figure 10 shows the mirror lines of Figure 9 mapped to the upper half plane H. Any other triangle in the tessalation is reached by applying a suitable element of PGL(2, Z), the group of integral 2ˆ2 matrices with determinant `1 or ´1 modulo t˘1u: This happens to be the Weyl group of our hyperbolic Kac-Moody algebra.It can be generated by the three reflections whose fixpoints form the three mirror curves mentioned above, which bound the fundamental triangle (4.8).The two generators of the even subgroup PSL(2, Z) are and its standard fundamental domain is cut in half by the extra reflection s 3 .In matrix representation p a b c d q we have up to multiplication by ´1 of course.The simple-root reflection s α0 : z Þ Ñ z 2z´1 appears in the middle of our ℓ"1 lists (4.3) and (4.7).Choosing it instead of s 3 leads to a fundamental domain with the cusp sitting at 1 rather than i8.In any case, it is clear that the potential U pzq " V `r"1, θpzq, ϕpzq ˘is a real automprphic function with respect to PGL(2, Z).We end this section by displaying log U in the Poincaré disk and in the upper half plane for the standard fundamental domain in Figure 11.

The potential as a Poincaré series
Our potential is a sum over all real roots α of AE 3 , thus each reflection s α inside PGL(2, Z) provides one summand.These reflections are given by traceless matrices R of determinant ´1,5  The scalar product in root space is an invariant bilinear form, α¨α 1 " trps α s α 1 q " 2 p p 1 `q r 1 `r q 1 P Z . (5.2) From (4.6) we see that the real function α¨x odd under s α becomes a real quadratic polynomial in z and z divided by |z´z|.A quick computation shows that α¨x " is indeed odd under the reflection (5.1).Therefore, on the upper half plane the potential is expressed as where z " x `i y and the sum runs over all reflections R in (5.1), with R and ´R contributing the same.

Comparison with
´α¨x " provides the translation between the labels pp, q, rq and pℓ, m, nq, up to a common sign of course.As an aside, we characterize in Table 1 the two orbits (e for ℓ even, o for ℓ odd) of the Weyl group on the set R of real roots.Restricting our potential to one of the two Weyl orbits removes either the ℓ"odd or the Hence, we can replace the sum over R with sums over appropriate orbits by the adjoint action of GL(2, Z) for a suitable reference, say and obtain 62 U pzq " (5.8) The question is: over which subset of GL(2, Z) matrices does this sum run?Since our fundamental domain F in (4.8) is half of the standard one for PSL(2, Z), all reflections R should be covered using adjoint orbits by matrices M n of determinant n " ad´bc equal to one or two.Indeed, for R P R `, we can find a unique (up to sign and the footnote) matrix M 2 such that R " M ´1 2 R 0 M 2 .In the case of R P R ´, in constrast, there are two such matrices M 2 , which have the form so such reflections R are covered twice by summing over M 2 .However, they also make up (again uniquely) the M 1 orbit of R 0 .Therefore, we can correct the overcount by subtracting, 2 U pzq " F 2 pzq ´F1 pzq for F n pzq :" 1 4 ÿ Mn upM zq . (5.10) The function F 2 can be obtained from F 1 by applying a Hecke operator T 2 (for weight k"0), and thus we have 2 U pzq " F 1 p2zq `F1 p z 2 q `F1 p z`1 2 q ´F1 pzq . (5.12) Therefore, it suffices to compute the Poincaré series upM zq . (

5.13)
There is another path to this result, which provides a useful connection to binary quadratic forms.7 Let us define [20] r where QpDq denotes the set of binary quadratic forms As 2 `Bst`Ct 2 over Z with discriminant D " B 2 ´4AC.For D"1 there is a bijection between Qp1q and PSL(2, Z) by solving We conclude that r F 1 " F 1 .For D"4 we have a bijection between Qp4q and our reflections R in (5.1), Therefore, the potential can also be expressed as (5.17) Now, Qp4q can be reduced to Qp1q, and it is not too hard to check that indeed [20] r F 4 pzq " pT 2 r F 1 qpzq ´r F 1 pzq " r F 1 p2zq `r F 1 p z 2 q `r F 1 p z`1 2 q ´r F 1 pzq , (5.18) confirming (5.12).It thus suffices to compute the generalized real-analytic Eisenstein series where Qp1q indicates a discriminant B 2 ´4AC " 1.As was shown in [20], this sum converges almost everywhere.8However, it does not decay at i8, but grows as r F 1 px`iyq " y 2 for y Ñ 8.
The form (5.17) can be translated back to the unit hyperboloid and indeed the Minkowski future, with the result where the sum runs over all integers A, B and C subject to the Qp4q condition B 2 ´4AC " 4. In this way, the real roots are parametrized by binary quadratic forms, which is of course equivalent to the solutions of the diophantine equation (2.8) but may be more convenient or manageable.

Dunkl operators
Calogero models in a Euclidean space are known to be maximally superintegrable.This is also the case for the spherical reduction of the rational models.One key instrument to establish this property is the linear Dunkl operators [21, 22] and their angular versions respectively.For rational models, the crucial property is the commutation rD i , D j s " 0, while the L ij deform the angular momentum algebra to a subalgebra of a rational Cherednik algebra [23].It is known that every Weyl-invariant polynomial in the D i or in the L ij will, upon its restriction 'res' to Weylinvariant functions, provide a conserved quantity, i.e. an operator which commutes with the Hamiltonian H or H Ω , respectively.Indeed, the Hamiltonians themselves can be expressed in this way, with the ground-state energy Let us repeat this construction for R 1,2 and the restriction to the hyperboloid x¨x " ´r2 " ´1.We follow the standard construction and define the 'hyperbolic Dunkl operators' (i " 1, 2) with α¨x " ´α0 x 0 `αi x i as deformed rotation and boost generators.Lorentz indices are raised and lowered with the Minkowski metric.In complex coordinates (4. which at g"0 reduces to the sop1, 2q algebra.The Opgq deformations are determined by the action of the differential parts on the reflection parts and by the commutators of the reflection parts themselves. A standard computation shows that the commutator of two linear Dunkl operators, rD µ , D ν s, reduces to the antisymmetric part (under µ Ø ν) of where the prime indicates excluding pairs with β"˘α.In the last step, under the sum we substituted β Ñ s α β " β ´pβ¨αq α, i.e. s β Ñ s α s β s α , and used s α α " ´α or s 2 α " 1.Hence, the criterion for linear Dunkl operators to commute is the vanishing of a two-form, where we abbreviated α µ dx µ "α and β ν dx ν "β.Note that the four pairs pα, ˘βq and p´α, ˘βq contribute equally to the double sum.
In order to generate the angular Hamiltonian H Ω in (1.4), we compute ÿ µăν L µν L µν " C 2 ´1 2 pB `B´`B´B`q " ´L2 `ÿ αPR x¨x pα¨xq 2 gpg´s α q `gSpgS`1q ´g2 η µν Y µν (6.11) by generalizing the results in [23] to R 1,2 .We remark that, due to the indefinite root-space signature and x¨x" ´1, the relative sign between ř L 2 and the angular Hamiltonian is flipped and the ground-state energy E 0 " ´1 2 res gSpgS`1q is negative and formally infinite.Besides this energy shift, our hyperbolic Dunkl operators can generate the angular Hamiltonian provided that pY µν q is not only symmetric but also traceless, i.e.
The commutativity of the linear Dunkl operators of rational Calogero models is a key property for their integrability in Euclidean space.It also assures the integrability of the angular models constructed by reduction to the sphere.It is therefore reasonable to perform this test also for our hyperbolic Kac-Moody Calogero model. 10We shall now investigate the conditions (6.10) and (6.12), i.e.Y " 0 and Y µ µ " 0. For classical root systems indeed Y " 0, because the double sum in (6.10) can be recast as a sum over planes of contributions stemming from the real root pairs lying in a given plane Π, which add up to zero for any such plane.In our hyperbolic model, this is obvious only for root pairs pα, βq at level ℓ"0, which form the A 2 subalgebra with a hexagon of roots and α¨β"˘1 throughout.Generically however, two arbitrary real roots α and β generate an infinite planar collection of real roots, .1) and their negatives.The roots in either string are related by hyperbolic reflections and rotations, but α and β need not be.All these comprise the real roots of a rank-2 subalgebra whose Cartan matrix reads [24] A m " and whose Weyl group is ␣ ps α s β q k´1 , ps α s β q k´1 s α ( for k P Z (7.3) because ps α s β q ´1 " s β s α and s 2 α " s 2 β " 1.Each odd element is a reflection on a hyperplane orthogonal to some real root γ k , while the even elements are elliptic, parabolic or hyperbolic elements of PSLp2, Zq, for mď1, m"2 or mě3, respectively.
Rather than finding the integral points on this quadric, we may compute the coefficients ξ and η recursively from (7.1).We recombine these two sequences in an alternating fashion (and flipping half of the signs) into a double-infinite sequence γ 2ℓ´1 " ps α s β q ℓ´1 α and γ 2ℓ " ps α s β q ℓ´1 s α β ô s γ k " ps α s β q k´1 s α ( with k, ℓ P Z. Due to s α α " ´α and s β β " ´β it reads and reproduces the ordering of the corresponding Weyl reflections in (7.3), with γ 0 " ´β and γ 1 " α.
The real roots γ k are the integral points on the positive branch of the quadric (7.4), while the negative branch contains the set t´γ k u.
Remembering s γ x " x ´px¨γq γ we combine the reflections and find for γ k " ξ k α `ηk β the recursion This yields the three-term recursion relation We note that the recursion can be iterated to the right as well as to the left, with ξ ´k " ´ξk and η ´k`1 " ´ηk`1 .(7.10) One may check that due to (7.9), so that indeed γ k ¨γk " ξ 1 ´ξ´1 " 2. The recursion can be solved explicitly, giving or via a generating function The zeros of the characteristic polynomial provide a simple closed expression, equally valid for both signs.Another useful parametrization of the real roots in Π m " xαβy is which exhibits the symmetry axis γ of the quadric (7.4).
Equipped with these tools, we can further specify with, representing Π m " xαβy, where we used that ξ k ξ k 1 ´1´ξ k 1 ξ k´1 does not change under a common shift of k and k 1 .
A more interesting case occurs for m"2.Here, the real roots lie on two straight lines (see Figure 13 left), where γ " α`β happens to be null and orthogonal to α and β.This set of roots creates the affine extension p sl 2 " p A 1 " A p1q 1 of sl 2 .Obviously, any pair of roots in this set has a scalar product of 2 or ´2.
due to cotpxq `cotpℓπ´xq " 0. Hence, also the affine subalgebras do not obstruct a commutativity of the linear Dunkl operators.
As soon as we go beyond level one, real root pairs with mą2 show up, and the associated quadric (7.4) is a hyperbola (see Figure 13 right for m"3).Let us inspect the simplest such case, m"3, where the coefficient sequence happens to be the even half of the Fibonacci sequence (k " 1, 2, 3, . ..), ξ k`1 " 3 ξ k ´ξk´1 with ξ 1 "1 & ξ 0 "0 ñ ξ k " 1, 3, 8, 21, 55, 144, 377, 987, . . ." f 2k , (7.24) where f n`1 " f n `fn´1 with f 0 " 0 [25,26].The root scalar products take the values ˘2, ˘3, ˘7, ˘18, ˘47 etc..In this case, the contribution to the two-form Y becomes with the understanding that f ´n " p´1q n`1 f n extends the Fibonacci sequence to the left.As numerical checks show, the individual sums over k P Z do not vanish, nor does the total expression.Turning off one of the two Weyl orbits in the root-sum for the Dunkl operator (6.1) does not help since odd values of m require both α and β to lie in R `, and thus only this orbit contributes here.We are forced to conclude that Y ‰ 0 for our model, so its linear Dunkl operators D µ do not commute to any simple expression, and so we cannot construct higher conserved charges in this way.Likewise, Y µ µ does not vanish either, and thus res ř L 2 in (6.11) does not reproduce the Hamiltonian H Ω .This points to a lack of integrability, but falls short of disproving it.

Conclusions
Spherical angular Calogero models are obtained by reducing a rational Calogero model in R n to the sphere S n´1 .Analogously, we have defined a hyperbolic angular model by reducing a Calogero Hamiltonian in R 1,n´1 to the (future) hyperboloid H n´1 .The main difference to the conventional angular model is the non-compactness of hyperbolic space and the replacement of a finite spherical Coxeter reflection group by an infinite hyperbolic one.As a consequence, the Calogero-type potential of the model is an infinite sum over all hyperbolic reflections and not easily obtained in a closed form.However, it is a real automorphic function of an associated hyperbolic Kac-Moody algebra.
We have worked out the details for the rank-3 case of AE 3 leading to a PGL(2,Z) invariant quantum mechanical model on the Poincaré disk or the complex upper half plane.In this case, the potential can be reformulated as a Poincaré series, which converges outside the mirror lines of PGL(2,Z).We then asked whether the integrability of the spherical angular models extends to the hyperbolic ones.To this end, we introduced the Dunkl operators for the AE 3 algebra, on R 1,2 and on H 2 , and computed their commutators.It turned out that the presence of hyperbolic rank-2 subalgebras in AE 3 prevents a simple result (like zero).This may be an obstacle to integrability, but is not enough to rule it out.
We comment that the energy spectrum of the AE 3 hyperbolic Calogero model is a deformation of the discrete parity-odd part of the spectrum of the hyperbolic Laplacian on square-integrable automorphic functions, because the singular lines of the potential impose Dirichlet boundary conditions on the boundary of the fundamental domain.It remains to be seen whether the spectral flow with the coupling g is isospectral under integral increments of g.
Since the Weyl-alcove walls of certain hyperbolic Kac-Moody algebras are the cushions of the billard dynamics in the BKL approach [27,28], the small-g limit of our hyperbolic Kac-Moody Calogero system provides a model for cosmological billards [13].The chaotic dynamics of such billards seems to be consistent with a formal integrability of the corresponding hyperbolic Toda-like theories [13].Furthermore, an alternative description of BKL dynamics leads instead to Euler-Calogero-Sutherland potentials of the sinh ´2 type, which also produce sharp walls in the BKL limit [29,30].This nurtures the hope that also our Calogero-type potentials retain a kind of integrability.Finally, the well-known quantum chaotic behavior of hyperbolic billards [31] may be "tamed" by turning on our Kac-Moody Calogero potential, since in the large-g limit the wave function will get pinned near the bottom of the potential.We hope that this opens a door to interesting further studies in this field.The very erratic dependence of the numbers U r pzq on r, due to the sum over square-roots of 1 modulo r in (A.6), prohibits further analytic simplification.For the same reason, the infinite sum for U pzq, although convergent, is not very tractable numerically.However, the convergence of the partial sums U pz, Rq for R Ñ 8 can be accelerated by adding a suitable correction term.The following heuristic argument suggests what this correction term should be.If the inner sum in (A.8) were over all values of p (mod r) then, since the value of a y 2 ´1 r 2 « y is close to y for r large, this inner sum for large r would simply be r times a Riemann sum for the integral ş R{Z Spx, yq dx " ş R px 2 `y2 q ´2dy " π{2y 3 and hence could be approximated by rπ{2y 3 .The actual inner sum is only over N prq rather than r values of p (mod r), where N prq denotes the number of square-roots of 1 modulo r.Hence, if these square-roots are more or less uniformly distributed on the interval r1, rs on the average, which is a reasonable heuristic assumption, then the value of the inner sum should be roughly N prq π{y 3 on average.Therefore, the contribution of the terms in (A.8) with rąR (the "tail") should be approximately π{y times ř rąR N prq{r 2 for R large.The value of the arithmetic function N prq fluctuates a lot, but its average behavior is quite regular, and one can give the asymptotic value of the sum ř rąR N prq{r 2 without difficulty.Specifically, from the Chinese remainder theorem we find that N prq is multiplicative, meaning that N `ś p νi i ˘" ś N pp νi i q, and N pp ν q in turn is easily evaluated as 2 for p an odd prime and νě1 (the only two square-roots of 1 in this case being ˘1 (mod p r )), and as 1 or 2 or 4 for p"2 and ν"1, 2, or ě3, respectively (the only square-roots of 1 in the latter case being ˘1 and ˘1`2 r´1 (mod 2 r )).This gives  numerically from our tabulated values (as we in fact did originally, with results that were not all that much worse than U p8q ).Alternatively, at the cost of a little loss of accuracy, one can replace U p0q pRq by the expression 2 U p0q pRq ´U p0q pR{2q, which eliminates any linear term in 1{R and is unchanged by employing U p1q , U p2q , or U p8q instead of U p0q .However, our analysis leading to the final correction term as given by (A.10), (A.13) and (A.20) with (A. 19) is mathematically interesting and seemed worth giving, especially in view of the unexpected occurrence of the nearly modular functions log ˇˇηpzq ˇˇand ℜ `E2 pzq ˘.

Figure 7 :
Figure 7: Contour lines of log V for the standard fundamental alcove intersecting the x 0 "1 plane

Figure 11 :
Figure 11: Contour lines of log U for a fundamental domain in the v disk (left) and the z plane (right)

ℓ m n p q r e o o o e o R `e e o o o e e o e e o o R ´o e e o e e Table 1 :
Assignment of Weyl orbit to a root αpℓ, m, nq or to a reflection Rpp, q, rq (e = even, o = odd) ℓ"even mirror lines from all diagrams and doubles the fundamental domain.In an Appendix with Don Zagier we outline how far one can proceed with an explicit computation of the potential function U pzq.The potential U pzq is a real modular function under the action of GL(2, Z), which is manifest viau R pM zq " u M ´1R M pzq for M z " az`b cz`dwith a, b, c, d P Z .(5.6)

Table 2 :
Rq 52.24167922 52.24327256 52.24429208 52.24465475 52.24484553 52.24496635 52.24500890 U pz2, Rq 52.24339662 52.24417862 52.24467954 52.24485793 52.24495185 52.24501138 52.24503236Values of truncated sums for the potential, at z 1 " 0.1 `0.7 i and at z 2 " ´1{z 1 "slicing by ℓ " formula (A.5).As a demonstration, we list in Table2ten-digit values of the partial sums U pz, Rq defined by truncating (A.8) at r"R for a typical point z 1 " 0.1 `0.7 i and values of R going up to one million.We have also included the values U pz 2 , Rq at the modular image z 2 " ´1{z 1 " ´0.2 `1.4 i, both as a test of the modularity of U pzq and as a confirmation of the accuracy of the computation.It takes PARI about 17 minutes for R"10 5 and about 28 hours for R"10 6 on a standard workstation to compute the values for each point z i given in this table.The output suggests that the final numbers are correct to about 7 significant digits.
psq :" ´1 `1 2 s `2 4 s `4 8 s `4 16 s `. ..¯ź pą2 ´1 `2 p s `2 p 2s `2 p 3s `. ..where ζpsq denotes the Riemann zeta function.In particular, N psq has a double pole at s"1 with principal part ζp2q ˘, so N prq behaves "on the average" like ζp2q ´1 log r, and ´1 log R R .This suggests that we can improve the convergence of ř U r pzq by replacing the partial sums U pz, Rq by U p0q pz, Rq " U pz, Rq This is indeed the case, as we see from Table3, in which we have tabulated the corrected partial sums with the same parameters as before.We can improve these values further by adding an appropriate term Rq 52.24419214 52.24462358 52.24488249 52.24496886 52.24501205 52.24503796 52.24504659 U p0q pz2, Rq 52.24465308 52.24485413 52.24497474 52.24501499 52.24503511 52.24504718 52.24505121 s´1

Table 3 :
Values of firstly improved truncated sums for the potential, at z 1 " 0.1 `0.7 i and z 2 " ´1{z 1